To determine the possible degrees of the polynomial based on the graph, we can observe the following:

1. Turning points: The number of turning points (local maxima or minima) helps to estimate the degree of the polynomial. In this graph, I see 3 turning points (two minima and one maximum).

2. Degree relationship: For a polynomial function of degree $n$, the maximum number of turning points is $n-1$. Therefore, if there are 3 turning points, the degree $n$ could be at least $4$ because $4-1 = 3$.

3. End behavior: The graph shows that both ends of the polynomial rise upwards as $x \to +\infty$ and $x \to -\infty$. This suggests that the leading term has an even degree because even-degree polynomials have matching end behaviors (both up or both down).

Possible Degrees:

Given that the graph has 3 turning points and the end behavior suggests an even degree, the possible degrees of the polynomial function are:

• 4
• 6
• 8

These are all even numbers greater than or equal to 4 (since 4 gives 3 turning points).

Would you like further details on the analysis or have any questions?

Follow-up questions:

1. What does the degree of a polynomial indicate about its shape?
2. How can you tell the degree of a polynomial from its graph alone?
3. What other features of the graph help identify the degree of the polynomial?
4. Can the degree of a polynomial be determined exactly from the graph, or are there multiple possibilities?
5. How do the coefficients of a polynomial affect the graph's shape?

Tip:

When examining the graph of a polynomial, always count the turning points and check the end behavior to estimate its degree.